Output Ripple Voltage for Buck Switching …

1 ApplicationReportSLVA630A January2014 RevisedOctober2014 OutputRippleVoltagefor BuckSwitchingRegulatorSurinderP. Singh, ,Manager, DesignCenterABSTRACTS witched-modepowersupplies(SMPSs)are usedto regulatevoltageto a ,whichcausesthe currentsand voltagesin the circuitto switchand outputvoltagealsohas rippleon top of the regulatedsteady-stateDC powersystemsconsiderthe outputvoltagerippleto be botha key parameterfor designconsiderationsand a keyfigureof onlineWEBENCH PowerDesignerrecognizesthe key importanceof peak-to-peakvoltageoutputripplevoltage the ripplevoltageis calculatedand reportedin the visualizer[1]. Thisapplicationreportpresentsa closed-formanalyticalformulationfor the outputvoltageripplewaveformandthe accurateoverall regionsof operationand harmonizesthe peak-to-peakripplevoltagecalculationover all regionsof newanalyticalformulationpresentedin this applicationreportgivesan accurateevaluationof the outputrippleas comparedto thesimplifiedlinearor root-meansquare(RMS)

2 This applicationreport,the analyticalmodelfor outputvoltagewaveformand peak-to-peakripplevoltagefor buckis ripplepeak-to-voltagefor variousinputconditionsandchoicesof outputcapacitorandcompareit reportanalyzesand presentsthe validityof the linearandthe TI portfolioare presentedthat comparethe experimentalripplewaveformsin Devicesand EvaluationModules(EVMs).. of the BuckSMPS:InductorCurrent,AverageOutputCu rrent,and CurrentThroughthe AnalyticalDerivationof the AnalyticalFormulationto TINA-TISPICES imulation, the AnalyticalFormulationto TINA-TISPICES imulation, the AnalyticalFormulationto TINA-TISPICES imulation,ESRof ..87 AnalyticalOutputWaveformfor ESRon Capacitanceon SwitchingFrequencyon January2014 RevisedOctober2014 OutputRippleVoltagefor BuckSwitchingRegulatorSubmitDocumentatio nFeedbackCopyright 2014,TexasInstrumentsIncorporated+ the Linearand RMSA pproximationAgainstthe AnalyticalFormulation, the Linearand RMSA pproximationAgainstthe the the the EVMof the systemunderconsiderationis a typicalbuckSMPS circuit(seeFigure1).

3 The switchingactionof thesynchronizedMOSFET causesthe currentin the inductorto havea triangularwaveform[2]. The DCcomponentof the inductorcurrentflowsthroughthe outputload,and the AC portionof the inductorflowsthroughthe outputcapacitor,as shownin Figure2. The time-varyingcurrentthroughthe capacitorcausesthe voltageacrossthe capacitorto be applicationreportaimsto presentan accurateyeteasy-to-implementformulationf or the BuckSMPSF igure2. Currentsin the BuckSMPS:InductorCurrent,AverageOutputCu rrent,and CurrentThroughthe OutputCapacitor2 OutputRippleVoltagefor BuckSwitchingRegulatorSLVA630A January2014 RevisedOctober2014 SubmitDocumentationFeedbackCopyright 2014,TexasInstrumentsIncorporated( )( )( )()tIIIC0i tv ti t Rdtvt0C=++= ( )( )( )RCv tvtvt=+( )( )p2pIonp2poffIti t1 22 TIt'i t'1 2II2T =- + =- ( )IonrippleIIonsw,,i (t)t0,Tit =i (t)tT ,T ( )( )0ripplei tIit=++ Iouti(t) derivethe outputvoltagewaveform,assumea simpleschematicof the outputcapacitor(seeFigure3).

4 The inductorcurrentis determinedby designparameterslike inputand outputvoltage,switchingfrequency,and is importantto notethe inductorcurrentis not dependenton the choiceofthe capacitor,and hencecan be consideredan outputcapacitorhascapacitance,C, and is assumedto havean equivalentseriesresistance(ESR), outputfilter of the SMPSis inductorcurrentis assumedto be an idealcurrentsourceunaffectedby the outputcapacitoror the Schematicfor AnalyticalDerivationof OutputVoltageWaveformThe currentflowingthroughthe inductor,as shownin Figure2 and Figure3, can be writtenas a sum ofthe averageoutputcurrentin the load and the ripplecurrent[2].(1)The ripplecurrenthas an averagevalueof 0 due to the charge-secondbalanceprinciple[2].

5 The ripplecurrentis composedof two piece-wiselinearanalyticalforms,as shownin Figure2. The first regioniswhenthe high-sideMOSFETis turnedon for timeTon, and the secondregionoccurswhenit is off for timeToff= Tsw Ton.(2)The total switchingtime isTsw= 1 / Fsw. Piece-wiselinearcurrentduringthe two portionscan be writtenintermsof the givenpeak-to-peakinductorcurrent,Ip2p:(3 )The originof the secondtime segmentfor analyticalis changedfor convenience:t' = t outputcapacitoris the sum of voltagedroppedacrossthe capacitance,C, andequivalentseriesresistance,R. Accordingto the Kirchhoff s VoltageLaw:(4)Usingthe relationshipof currentto voltageacrossthe capacitor:(5)3 SLVA630A January2014 RevisedOctober2014 OutputRippleVoltagefor BuckSwitchingRegulatorSubmitDocumentatio nFeedbackCopyright 2014,TexasInstrumentsIncorporated()( )222p2pp2poffout, maxIImaxoffoffIR CITvv t 'RCT2CT2 ==+- offmaxTt 'RC2=-( )2p2pp2pIIoffoffIIt'v t' R1 2t'2T2CT =-+- t'()( )222p2pp2ponout,minIminononIR CITvv tRCT2CT2 == --- onminTtRC2=-()Imindv t0dt=( )( )( )tIII0tp2pp2p0onon2p2pp2pononi tv t i t RdtCIIttR1 21 22T2 CTIIttR1 2t2T2CT=+ =- ++- + =- ++- + loss of generality,assumethe initialvoltageon the capacitoris 0.

6 Use the valuesfromEquation3in Equation5 to get:(6)Equation6 is quadraticin time;hence,the outputvoltagewaveformhas a voltagereachesits minimumatt = tmin, whichis obtainedby solvingthe followingequation:(7)Solvingfortminby usingEquation6 and Equation7 we obtain:(8)Equation8 statesthat the minimumis shiftedfromcenterof the on time periodTonby the RC minimumtime cannotbe < 0, it impliesthattmin minimumvoltageattminis:(9)Similarly,for the secondtime segment,outputvoltageis:(10)Whosemaximum voltageoccursat:(11)Also,becausethis time cannotbe negative,t'max 0. The maximumvoltageis:(12)This ,the waveformof the outputrippleis givenbyEquation6 and Equation10. Second,the maximumoutputvoltageis givenby Equation12, and theminimumoutputvoltageis givenby Equation9.

7 Now,the peak-to-peakripplevoltagein variousregimescan be BuckSwitchingRegulatorSLVA630A January2014 RevisedOctober2014 SubmitDocumentationFeedbackCopyright 2014,TexasInstrumentsIncorporated()swout , p2pCRRCFVV V2D 1 D =+ - ()2p2pp2pswout, p2pswII R CFV8CF2 D 1 D=+ -()()onswswoffswswon offswswswonoffDT DTF1 DT1 D TF1T T TFF1 1T T D 1 D==-=-=+==+=-( )( )222222p2pp2pp2pp2poffonout, p2poffoffononIR CIIR CITTVRCRCT2CT2T2CT2 =+-++- onoffminmaxTTtRC 0, and t 'RC022=->=->()()out, p2pIImaxIminVvt 'v t= outputvoltagepeak-to-peakrippleis the differencebetweenthe maximumand minimumof thewaveform:(13)The analyticalexpressionsare givenin threepossibleregimesfor the outputcapacitorresistance-capacitancetim e constant,RC: small,intermediate,and capacitorRC productis small:(14)ApplyingEquation13, the peak-to-peakoutputvoltageis:(15)Notingth at:(16)Arriveat the expressionfor outputpeak-to-peakripplevoltage:(17)Ofte n,engineersthinkin termsof contributionto outputrippledue to the capacitoronly,or due capacitor-onlyportionis obtainedby settingequivalentseriesresistanceto 0,R= 0,whichgivesthe familiarexpressionVC= Ip2p/ (8 CFsw).

8 If the resistorwas aloneon the outputcircuit,thenthe resistor-onlyportionisVR= Ip2pR. Equation17 can be writtenas:(18)ComparingEquation18 with Equation17 revealsthe total outputripplecannotbe writtenin termsof asimplesum of rippledue to the capacitoror resistoralone;otherfactorssuchas duty cycle,switchingfrequency,and the valuesof resistorand capacitor,also play a providedin thediscussionof January2014 RevisedOctober2014 OutputRippleVoltagefor BuckSwitchingRegulatorSubmitDocumentatio nFeedbackCopyright 2014,TexasInstrumentsIncorporated( )222p2pp2pp2ponout, p2pononIRIR CITVRC2T2CT2 =++- onoffTTWhen RC, and RC22<>( )222p2pp2pp2poffout, p2poffoffIR CIIRTVRCT2CT22 =+-+ onoffTTWhen RC, and RC22> <onoffTTRC, and RC22<>onoffTTRC, and RC22> <()()p2pp2pout, p2pIImaxIminp2pIRIRVv0v t0t 'IR22 ==-==- -= onoffTTRC, and RC22>> large.

9 Usetmin= 0 andt'max= 0 in Equation6 and Equation10 to calculatethe peak-to-peakoutputvoltage:(19)Equation19 impliesthat for a largeESRoutput,voltagerippleis independentof also independentof othervariablessuchas duty cycleand of an intermediatevalue,it is either:(20)Or(21)One extremeof the waveform,eitherthe maximumor minimumvoltage,is dependentonly on peak-to-peakcurrentand otherside mustbe estimatedfromeitherEquation9 orEquation12.(22)(23)6 OutputRippleVoltagefor BuckSwitchingRegulatorSLVA630A January2014 RevisedOctober2014 SubmitDocumentationFeedbackCopyright 2014,TexasInstrumentsIncorporatedD = , C = 10 F, R = 0 , Ip2p = 2 AFsw = 125 kHz!D = , C = 10 F, R = 0 , Ip2p = 2 AFsw = 125 kHz!

10 Analyticalformulationis validatedagainstSPICETINA-TIsimulations[ 3]. Comparefor , showsthe simplecaseof a duty cycleof and an ESRof 0. SPICE waveformand theanalyticalformulationagreewith waveformis quadraticin time in bothof the quadraticshapeis sometimesmistakenas nonsinusoidalquadraticnatureis evidentfor a duty cycleless thanor , as shownin Figure5, wherewe comparethe analyticalformulationagainstSPICETINA-TI ,for a duty cycleof set to 0, so the curvesshowa fullyquadraticformulationin usesthe sameconditionsexceptthat ESRis ,whichmakesRC > Ton. The waveformshapehas a sharptriangular-likeshapeduringthe on ripplewaveformoccursat time 0. The analyticalformulationagreeswith SPICETINA-TIfor 125 kHz at duty cycleof with Iout peak-to-peakof 2 10 F with an ESRof 0.